JMathPhys_34_969

Quantum group symmetric Bargmann-Fock space: Integral kernels, Green functions, driving forces Achim Kempf Sektion Physik der Ludwig-Maximilians-Universitiit, Theresienstr.37, 8000 Miinchen 2, Germany (Received 2 September 1992; accepted for publication 20 December 1992) Raising and lowering operators that transform under the SU,(n)-quantum group get deformed commutation relations. They are represented as adjoint operators on a Hilbert space of noncommutative holomorphic functions. Through the algebraically defined integral on this function space, every operator on the Fock space can also be represented as an integral kernel. The Green function for free harmonic oscillators and spin-1/2’s in a constant magnetic field is given. Further on it is studied how such systems react on the switching on of a driving force. Calculating the vacuum-vacuum transition amplitude, it is found that the deviations from the undeformed case grow with the strength of the driving force. Throughout all calculations the bosonic and the fermionic cases are considered simultaneously. I. INTRODUCTION Quantum groups originate in physics (quantum inverse scattering method), although the underlying mathematical concept of Hopf algebra is of course much older. They have a wide field of applications from statistical mechanics to knot theory and are today also studied in their own right, motivated by their fundamental character, being a deformation of groups (see e.g., Refs. 1, 2, and references therein), In particular the development of a differential calculus on quantum planes3 was the starting point for the present work. In the following the intention is, as a new application of quantum groups to physics, to implement the quantum plane deformation in quantum mechanical commutation rules. Apart from this I will however try to stay as close as possible to usual quantum mechanics. The main concepts, Hilbert space of states and unitarity of time evolution, will be preserved. Usually in quantum mechanics Euclidean vectors of position and momentum generate the Heisenberg algebra, which is then represented on the Hilbert space of square integrable func- tions. However one may as well generate the Heisenberg algebra by the raising and lowering operators, so that it is then natural to represent the algebra on a space of holomorphic func- tions. This has, e.g., the advantage to exhibit for the n-dimensional harmonic oscillators and n spin-l/2 systems their SU(n) symmetry, which is a larger symmetry group than SO(n). Here the basic objects are chosen to be the raising and lowering operators which shall transform under the SU,(n)-quantum group. In Sec. II the deformed commutation relations for the fermionic and bosonic algebras will be derived. There are already several approaches to q-deformed algebras of raising and lowering operators in the literature (see, e.g., Refs. 4-7). The bosonic algebra that 1 obtain was first found in Ref. 8 (as I only learned recently). In Sec. III, fermionic and bosonic raising and lowering operators are represented as adjoint operators on a Hilbert space of deformed holomorphic functions, i.e., on a SU,(n)-symmetric Bargmann-Fock space. The scalar product on this space is given by an algebraically defined integral. Using this integral it is shown in Sec. IV how to represent arbitrary operators as integral kernels on the function space. As an application in Sec. V simple systems are studied, as such with driving forces. The vacuum-vacuum transition amplitude for the switching on of a constant force is calculated J. Math. Phys. 34 (3), March 1993 0022-2488/93/000969-l 9$06.00 @ 1993 American Institute of Physics 969 Downloaded 13 Apr 2011 to 202.115.51.3. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions 970 Achim Kempf: SU,(n)-symmetric Bargmann-Fock construction explicitly for the leading orders in the driving force. One finds that deviations from the unde- formed case do not occur before the second order. While much of the preparatory Sets. II and III are already contained in previous work by the author,g the rest is new. Since we will work with the quantum group SU,(n), let us collect some of its properties - - and notations (see, e.g., Ref. 1). The R matrix read; R=qzejsej+ c ej@d+(q-l/q) z e>@d, i i#i i>j with e$Mn( C) matrix units. It fulfills the quantum Yang Baxter equation (QYBE) R;&J+R!$@& which is also written One has additionally and the Hecke relation where RnRdb=R23RnRn. qdRf and RF;= R$ -$+(b+f)~+l=o, . . . R’J* =Rf, and kl’ b:= -l/q, f:=q. (1) (2) (3) (4) (5) (6) We denote the generators of Fun(SU,(n)) by Uij and S is the antipode. An involution is defined by*:+S(rrj). For later convenience let us generalize the notation that was used in Eq. (3) to abbreviate the notation in Eq. (2). In Eq. (3) all index numbers are written in lower position. Equal index numbers indicate a summation from bottom left to top right in Eq. (2). The free indices at the beginning and the end of such a summation are the same on the left-hand side and right-hand side of the equation. For example, the index number 2 indicates the j summation in Eq. (2). The free indices corresponding to index number 2 are b and e. Let us now extend this notation for index numbers in upper position: They shall indicate summations from top left to bottom right. Thus, for example, z&!?( z~f) R$= R$S( uy) u6 is u,S(U~)R~~=R,~S(U~)U~ or u1R12S(u2) =S(u2Mw . (7) Note that the permutation of two factors (if they commute) implies the raising or lowering of the index numbers that they have in common. The comodule action for contravariant and covariant quantum planes [which are Fun(SU,(n))-comodule algebras21 with coordinates xi and yi (i= l,...,n) reads in this notation &ci=ui @xl and S$=S(u’) ey’. J. Math. Phys., Vol. 34, No. 3, March 1993 Downloaded 13 Apr 2011 to 202.115.51.3. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions Achim Kempf: SU,(n)-symmetric Bargmann-Fock construction 971 Formal Hermitian conjugation +: xi + xf maps contravariant planes into covariant planes (and vice versa), e.g., sxt=ufj~x5=u~~~x~=s(ui) @Xj. (8) It is well-known that on contravariant quantum planes the following commutation relations are preserved under the comodule action (see, e.g., Ref. 2) and also that due to the QYBE no new relations of higher degree follow. The Hermitian conjugate quantum plane has the commutation relations [we used Eq. (4)] which transform covariantly. From the Hecke relation it follows that only two choices of c lead to nontrivial commu- tation relations. For c= b we are working with an deformed antisymmetrizer, in the case c= f it is a deformed symmetrizer. The PoincarC series of the quantum plane coincides in both cases with the undeformed ones.’ In particular the symmetrized quantum planes are finite dimen- sional. Let us call the case c = b bosonic and the case c = f fermionic. II. FOCK SPACES Let us introduce a covariant quantum plane of “raising operators” Uti and a contravariant plane of “lowering operators” a’. A. Commutation relations The commutation relations at’at2+cR2&$=0 , a1~2+cR21qa1=0, follow immediately and for the mixed relations we make the ansatz alat2+Q12at2al=S2 1. Using the commutation relations and that the algebra is associative one finds that a,atzat3 = - Q12a+*alat3 +at3S; = Q12Q13at2at3al - Q126:at2 +at3@ = -cQ12Q13R32a+3at2al - Q126;a+“+at36f equals (9) (10) J. Math. Phys., Vol. 34, No. 3, March 1993 Downloaded 13 Apr 2011 to 202.115.51.3. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions 972 Achim Kempf: SU&)-symmetric Bargmann-Fock construction In the undeformed case the terms of the third lines that are cubic and those that are linear in the generators coincide separately. Since we do not want to change things drastically we require the same to hold in the deformed case too. Actually there is a natural choice for Q such that this implies no new relations among the generators, If we choose Qz= I /CR the relations and are simply the QYBE and the Hecke relation. Thus the following choice of mixed commutation relations is natural ala?+ l/~R~~a+~a, =S2 1. (11) In the subsequent sections further commutation relations between generators of different quan- tum planes will also be determined by this naturalness consideration (else the ideals that we divide out of the free algebra in order to implement the commutation relations might be too big, making the resulting algebra trivial). It remains to check that the relations are preserved under transformation [we used Eq. (7)]. Formal Hermitian conjugation is consistent because, using Eq. (4) B. Number operator and norm on Fock space There is a scalar transforming and Hermitian number operator N: = atpi. Its commutation relations are derived from al(at2a3) = - I/cR12at2ala3+Sfa3= l/$R12R-‘13a~a3al+Sfa3 . Contracting index numbers 2 and 3 we arrive at aiN- 1/2Na’=a’ and by Hermitian conjugation Nati- l/c2atfl=at. 1 a Let us define the normalized ground state as usual (12) (13) (O/O)=1 and a’(O)=0 for i=I,...,n. J. Math. Phys., Vol. 34, No. 3, March 1993 Downloaded 13 Apr 2011 to 202.115.51.3. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions Achim Kempf: SU,(n)-symmetric Bargmann-Fock construction 973 By successive application of at’s on IO) the Fock space is spanned. Treating the a’s and at’s as adjoint operators, a scalar product (,) is defined inductively. It will be studied in detail in the following section. Let us convince ourselves that all states have positive norm (even without further restric- tions on q, which was not known in Ref. 9). In order to calculate we observe, that (not summed over i) and ajar= 1- l /cR$a$P= 1- l/c(qa{a’+ (q- l/q) c a’p’), j 1. In the latter case Ep eventually converges towards w/( 1 -c-‘) and H is a bounded operator. One could then choose o < 0 even for bosonic and infinite fermionic systems. Further on, since there is then an upper bound to the uncertainty in energy, we expect a minimal uncertainty in time (18) Nevertheless, we stick to the undeformed Schradinger equation i( d/dt) \I’( t) =HY (t) so that time is preserved as a real parameter and the time evolution remains unitary. One may now proceed with an examination of (Hermitian) position and momentum operators (in the notation for the bosonic case) (19) In particular their spectrum, eigenstates, and transformation properties would be of interest. Furthermore, e.g., anharmonic oscillators could be considered. Let us however proceed with the formalism. As a first step we will represent the a’s and ah on a space of holomorphic functions. For the undeformed case see, e.g., Ref. IO. III. BARGMANN FOCK REPRESENTATION In the following the operators a and at shall be represented as differentiation and multi- plication operators on a space of (deformed) holomorphic functions p: fi-dq’ p: a+idfji. The complex conjugation map, denoted by a bar, is decked as follows: For the complex coefficients of the polynomials the bar operation means ordinary complex conjugation. Let us call polynomials exclusively in ?j’s holomorphic. The bar we define to be an antialgebra mapping, i.e., we have, for example, A. Commutation relations a,iii,= $ani . From the algebra of the raising and lowering operators and by complex conjugation we get immediately $fj2+cR2’~2if1=0, 77,~2+cR2,r/2rj, =O, (20) aij,aii,+CR2,aii2aii, =o, a,1a,2+CPa,2a,1=0, a+$+ l/~R,~$a~, =S;, q,a,2+ l/~R~~a,~r]~ =Sf. (22) In Ref. 9 it was shown that several choices of natural mixed commutation relations between the barred and unbarred entities are possible (natural in the sense of Sec. II A). It was also found that the following choice of relations allows a new and unified prescription for bosonic and fermionic integration, i.e., for the scalar product on the Hilbert space of holomorphic functions J. Math. Phys., Vol. 34, No. 3, March 1993 Downloaded 13 Apr 2011 to 202.115.51.3. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions Achim Kempf: S&,(n)-symmetric Bargmann-Fock construction 7gj2+i/~R,2ji27j,=o, aq,a~2+cR-12,a~2aq,=o, $aq2+ ihzPa,2ij~=o, aii,q2+ i/cR2,q2aq, =o. It follows that the following c-deformed exponential function (a,a+$ ec(aqp,-‘L= c I C2r- 1 r [TIC? with [r] C: =.m has the commutation relations ecca$%‘) fja = fg,($,+) + a,aec(dqidqi) and complex conjugated 975 (23) (24) (25) (26) a rl ec (a,,+) = ec(aq,+i)qa + ecC3q,&ji)a,ia. (27) The evaluation of differentiation is defined such that it commutes with transformations and complex conjugation: All d,‘s are to be commuted to the right, the dV’s to the left. When they arrive, the corresponding terms are to be set equal to zero. What remains is the value of the differentiation. B. Hilbert space structure The scalar product (, ) for holomorphic polynomials is given’ by the following “integral:” (dvJ):= (&‘er(a@~)+),vduatd at ll=~=q. (28) The 4 and $ are arbitrary polynomials in ?j’s with complex coefficients. Evaluation at 7 = 0= ?j means that at first the differentiations are to be evaluated. Then all terms containing 71’s and q’s are to be set equal to zero. Thus the result is a number. The procedure is transformation invariant because transformations map derivatives onto derivatives, coordinate functions onto coordinate functions, and numbers onto numbers (which form a singlet representation). The transformation invariance of the “integral measure” e, (%aGi) is clear and also it is obvious that (,) is a Hermitian form. The operators ak and al are adjoint in respect to (,) - (p(a+kMtCI)= (rlk&, ‘d~f+)$L”,l. at 0 = ( fj?$e,‘a+i?,i)@)evd, at o = (f$ee,(a+%%+~)e.d. at o+ (&@d~)a~k~),va~, at o =(hp(ak)ll/). The wave function of the ground state is 1. It is normalized (1,l) = (e,(a&ii))evaluated at o= 1. Thus the scalar product (,) coincides with the bracket (,) of the Fock space. In particular from Sec. II B is known that it is positively definite. J. Math. Phys., Vol. 34, No. 3, March 1993 Downloaded 13 Apr 2011 to 202.115.51.3. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions 976 Achim Kempf: S&,(n)-symmetric Bargmann-Fock construction The Hilbert space of all wave functions is defined to be the set of power series in ?j’s that are square integrable in respect to (,). Let us introduce a more familiar notation for the scalar product s dfj dq $ep%‘q!x= ($ejaqPi”?j),aluated at ,,=o=q . Note that the evaluation procedure for the above defined integral is the same in the bosonic as in the fermionic case while one usually uses complex integration in the one and Berezin integration in the other case. IV. INTEGRAL KERNELS Like in the undeformed case it is possible to represent every operator P (expressed in at and a or equivalently in q and a,) as an integral kernel. Once the operator P is normal ordered, there is a simple rule how to get its integral kernel Gp which is a function of ;ii’ and q. Integrating any wave function $(?j) over GP(+j’,v) leads then to a function of ij’, which is Y(T) dfj dq Gp( ij’,q)e~~‘$(fj) =p~cl(q’). (30) To this end the commutation relations between the two copies (primed and unprimed) of the function space have to be specified. A. Commutation relations between copies of the function space The following ansatz leads to a simple formula for the integral kernel of the identity operator: (fihi)agk -a,j(+fj =q;, (31) Thus we set (32) so that from our ansatz is reproduced after contraction of the index numbers 1 and 2. The relations r]1fj’2+cR-‘21fj’2qI=0 (33) are natural in the sense of Sec. II A, which is seen when considering the different ways to write r]‘@ & by using the commutation relations. Analogously, considering the terms aFir& and aYi&,,ii;, and by application of the natu- ralness condition the following relations follow: aii’,q2+cR-1,2q2aiit,- -0, aiif,avZ+ ih~~~a,~a,,,=o. Then, contracting in (34) J. Math. Phys., Vol. 34, No. 3, March 1993 Downloaded 13 Apr 2011 to 202.115.51.3. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions Achim Kempf: SU,$n)-symmetric Bargmann-Fock construction (,7’1,,2)~3= --CR-‘32ij”75’3772=C2R-‘32R31~3~l~2 the index numbers 1 and 2 yields (ij;Tji,~~-c2Tjpj;?j) =o. Thus follows 977 (35) (~~~‘)‘d,k=d,k(~:77’)‘-t(l+Cz+...+C 2+19ij;(jri’7?i)‘-l and e(ii:?7i) c ++pi) +ij;pye (36) B. Integral kernel of the identity operator Actually e:j”) is the integral kernel of the identity operator. We prove this by induction, noting first that it is mapping the ground state onto itself I &j dv ,(vld),(a,~i) 1_ 1 - . c c To complete the induction the assertion shall now hold for a wave function I/J(~). We show that ,lii;tl’) maps q&(q) onto itself too s dq-j dq ,m;?‘p,d!) - c c qktj(+= dfdq e111’7i’(iik+a?*)~~S’3(ii) s = s &j drl e~ii:~i)+$-%+$(~) = s &j d7](ii;+a,k)ek’i;ni)e~~)~(~) =q; s &j d,, ej”:“i’e~;‘$,(q) =fGKT). Note that in the undeformed case the integral kernel G, of the identity is ,(qj’J’). C. Completing the set of commutation relations Equations (33)) (34)) and their complex conjugates are not yet the complete set of com- mutation relations between the primed and the unprimed entities. The remaining relations follow from the requirement s dfj’ dr]’ G, (fj”,f)e,@fr+‘)G, (fj’,~) = GI (fj”,v), (37) i.e., J. Math. Phys., Vol. 34, No. 3, March 1993 Downloaded 13 Apr 2011 to 202.115.51.3. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions 978 Achim Kempf: SU&)-symmetric Bargmann-Fock construction s dfj’dq’ ec I e c . (75:‘s’i)ec(ada~,i)e(ii;?i) =,(f$+) Thus the following equation must hold: s dij’ d??‘(~~,‘i)r(a,,ia.i)r(~~~i)r=(~~?7i)’( [&!)2. (38) This can only be achieved with the following ansatz of commutation relations: 71;772+cR-‘12712d=O. (39) Considering the different ways to write the terms 7;r/2&,,3, vlr];JV3, and ~lL9B12d,3 one finds that only the following choice of commutation rela